Contemporary Mathematics
Volume 178, 1994
On h-Vectors and Symmetry
Ron M. Adin
Abstract
Tight lower bounds are obtained for all the face-numbers of a rational simpli-
cial polytope that admits a fixed-point-free linear symmetry of prime-power
order. These bounds are best possible for all compatible values of the poly-
tope's dimension, its number of vertices, and the order of its given symmetry.
They are consequences of corresponding bounds on the h-vector of the poly-
tope, and extend previous results of Stanley and of the author. Also included
is a short survey of some concepts and results in the combinatorial theory of
group actions on polytopes.
1 Statement of Results
The purpose of this paper is, primarily, to give the details of a certain result about
polytopes with a given symmetry. We also take this opportunity to include a brief
survey of terminology and results concerning group actions on polytopes. Therefore,
this opening section contains complete statements of the main results, whereas all
the background material, including definitions of some terms used in the statement
of the results, is deferred to Section 2. Details about the organization of the rest of
the paper may be found at the end of the current section.
Theorem 1
Let P be
a
simplicial
convex
d-polytope with rational vertices, admitting
a
fixed-
point-free linear action of
a
cyclic group G of order n, where n
=
p" is
a
prime
power. The dimension d is necessarily divisible by ¢(
n) =
p"-
1
(p - 1), where ¢
is Euler's totient function. Let
Pmin(G,
d) be the free sum of dj(p-
1)
copies of
the (p- 1)-simplex, and let the difference between the h-polynomials of P and of
Pmin(G,
d) be
d
hp(q)- hpmin(G,d)(q)
= L:iqi.
(1)
i=O
1991 Mathematics Subject Classification. Primary 52B05; Secondary 05E25, 14M25, 52B15,
52B20.
Research supported in part by the Israel Science Foundation, administered by the Israel
Academy of Sciences and Humanities.
1
©
1994 American Mathematical Society
0271-4132/94 $1.00
+
$.25 per page
http://dx.doi.org/10.1090/conm/178/01889
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